A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888

Offset: 1

Author

Gus Wiseman, Jul 12 2023

Keywords

• nonn

Comments

Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Examples

			The terms together with their prime factors begin:
    2 = 2
    4 = 2*2
    8 = 2*2*2
   16 = 2*2*2*2
   18 = 2*3*3
   32 = 2*2*2*2*2
   50 = 2*5*5
   54 = 2*3*3*3
   64 = 2*2*2*2*2*2
   98 = 2*7*7
  108 = 2*2*3*3*3
  128 = 2*2*2*2*2*2*2
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

For any unique co-mode: A359178, counted by A362610, complement A362606.

For high mode: A360013, positions of 1's in A363487, counted by A241131.

For low mode: A360015, positions of 1's in A363486, counted by A241131.

Partitions of this type are counted by A364062.

For low co-mode: A364158, positions of 1's in A364192, counted by A364159.

Positions of 1's in A364191, high A364192.

A112798 lists prime indices, length A001222, sum A056239.

A356862 ranks partitions w/ unique mode, count A362608, complement A362605.

A362611 counts modes in prime indices, triangle A362614.

A362613 counts co-modes in prime indices, triangle A362615.

Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.

Programs

Maple

filter:= proc(n) local F,F2,Fo;
  F:= ifactors(n)[2];
  F2,Fo:= selectremove(t -> t[1]=2, F);
  Fo = [] or F2[1,2] < min(Fo[..,2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Select[Range[100],comodes[prifacs[#]]=={2}&]

Python

from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(),default=0) or m==1, count(max(startvalue+startvalue&1,2),2))
A364061_list = list(islice(A364061_gen(),30)) # Chai Wah Wu, Jul 14 2023

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - Amiram Eldar, Aug 30 2024